In conventional traveling-wire electroerosion machines widely in production use today, the electrode wire is arranged to travel vertically up to down or vice versa and the work supporting plane is arranged to extend horizontally. As a result, when the traveling electrode wire finishes cutting over the entire closed-loop path, the central severed portion of the workpiece becomes unsupported and, due to gravity drops off the supported peripheral portion, tending during drop to hit on the wire and hence to give rise to wire breakage and to damaging short-circuiting against the electrode wire. While various measures such as the use of a permanent magnet designed to magnetically bridge the two portions before they are separated have been used to overcome the problem of drop-off of the central severed portion, the conventional measures require an interruption of the machining operation and have been found to be inefficient and unsatisfactory.